Computing power series in polynomial time
Advances in Applied Mathematics
Complexity theory of real functions
Complexity theory of real functions
Reachability analysis of dynamical systems having piecewise-constant derivatives
Theoretical Computer Science - Special issue on hybrid systems
Complexity and real computation
Complexity and real computation
A computational model for metric spaces
Theoretical Computer Science
Effective properties of sets and functions in metric spaces with computability structure
Theoretical Computer Science - Special issue on computability and complexity in analysis
Model checking
Computable analysis: an introduction
Computable analysis: an introduction
A new kind of science
Effective metric spaces and representations of the reals
Theoretical Computer Science
Perturbed Turing Machines and Hybrid Systems
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
Uniform test of algorithmic randomness over a general space
Theoretical Computer Science
Constructing non-computable Julia sets
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Computability of Julia Sets
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
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Computation plays a key role in predicting and analyzing natural phenomena. There are two fundamental barriers to our ability to computationally understand the long-term behavior of a dynamical system that describes a natural process. The first one is unaccounted-for errors, which may make the system unpredictable beyond a very limited time horizon. This is especially true for chaotic systems, where a small change in the initial conditions may cause a dramatic shift in the trajectories. The second one is Turing-completeness. By the undecidability of the Halting Problem, the long-term prospects of a system that can simulate a Turing Machine cannot be determined computationally. We investigate the interplay between these two forces -- unaccounted-for errors and Turing-completeness. We show that the introduction of even a small amount of noise into a dynamical system is sufficient to "destroy" Turing-completeness, and to make the system's long-term behavior computationally predictable. On a more technical level, we deal with long-term statistical properties of dynamical systems, as described by invariant measures. We show that while there are simple dynamical systems for which the invariant measures are non-computable, perturbing such systems makes the invariant measures efficiently computable. Thus, noise that makes the short term behavior of the system harder to predict, may make its long term statistical behavior computationally tractable. We also obtain some insight into the computational complexity of predicting systems affected by random noise.